A new interpretation of the misadjustment in adaptive filtering

An LMS type adaptive filter in a stationary environment does not converge to the Wiener solution, but exhibits small asymptotic fluctuations around it. These contribute to the output as an interfering signal, whose strength is conventionally designated by the term “misadjustment”, defined as the normalized excess mean-squared error due to the weight fluctuations. Importantly, this definition makes sense only for a white reference signal. For a coloured reference signal the contribution of the weight fluctuations to the output signal is often correlated with the reference signal which forbids direct superposition of the partial powers. In that case the excess mean-squared error no longer serves as a measure of the deviation between the error signal and the reference signal. In a small-signal approximation the power of the partial output signal due to the weight fluctuations is then shown to represent an appropriate measure of the misadjustment